3/07/2020
We decided to do central Colombia, basically because it is where the capital is.
We built a model for the number of confirmed cases using all the others covariates (plus some we created) and we estimated the predictive accuracy of our selected model.
We decided to consider as central Colombia the following cities:
Here we can see our selected cities. The color of the pins is related with the number of cases: if they are less than \(10\) the color is “green”, if they are less than \(100\) the color is “orange”, otherwise it is “red”.
## ID de caso Fecha de diagnóstico Ciudad de ubicación Departamento o Distrito ## 1 1 2020-03-06 Bogotá Bogotá D.C. ## 2 2 2020-03-09 Buga Valle del Cauca ## 3 3 2020-03-09 MedellÃn Antioquia ## 4 4 2020-03-11 MedellÃn Antioquia ## 5 5 2020-03-11 MedellÃn Antioquia ## 6 6 2020-03-11 Itagüà Antioquia ## Atención Edad Sexo Tipo PaÃs de procedencia Grupo de edad ## 1 Recuperado 19 F Importado Italia 19-30 ## 2 Recuperado 34 M Importado España 31-45 ## 3 Recuperado 50 F Importado España 46-60 ## 4 Recuperado 55 M Relacionado Colombia 46-60 ## 5 Recuperado 25 M Relacionado Colombia 19-30 ## 6 Casa 27 F Relacionado Colombia 19-30 ## Continente de procedencia ## 1 Europa ## 2 Europa ## 3 Europa ## 4 Colombia ## 5 Colombia ## 6 Colombia
## Date Elapsed time Department New cases/day Cumulative cases/Department ## 2 2020-03-09 3 Antioquia 1 1 ## 4 2020-03-11 5 Antioquia 3 4 ## 9 2020-03-14 8 Antioquia 3 7 ## 11 2020-03-15 9 Antioquia 1 8 ## 24 2020-03-19 13 Antioquia 3 11 ## 28 2020-03-20 14 Antioquia 11 22
Number of cases confirmed day by day
Something.
As a first attempt, we fit a simple Poisson regression:
\[ ln\lambda_i = \alpha + \beta\cdot elapsed\_time_i \\ y_i \sim \mathcal{Poisson}(\lambda_i) \] with \(i = 1,\dots,83\), and \(y_i\) represents the number of cases.
## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -I"/usr/share/R/include" -DNDEBUG -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/Rcpp/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/unsupported" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/BH/include" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/src/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/rstan/include" -DEIGEN_NO_DEBUG -D_REENTRANT -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp -fpic -g -O2 -fdebug-prefix-map=/build/r-base-V28x5H/r-base-3.6.3=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Core:88:0,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Dense:1,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4,
## from <command-line>:0:
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name ‘namespace’
## namespace Eigen {
## ^~~~~~~~~
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## namespace Eigen {
## ^
## In file included from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Dense:1:0,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4,
## from <command-line>:0:
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: File o directory non esistente
## #include <complex>
## ^~~~~~~~~
## compilation terminated.
## /usr/lib/R/etc/Makeconf:168: recipe for target 'foo.o' failed
## make: *** [foo.o] Error 1
## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 1). ## Chain 1: ## Chain 1: Gradient evaluation took 2.2e-05 seconds ## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds. ## Chain 1: Adjust your expectations accordingly! ## Chain 1: ## Chain 1: ## Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 1: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 1: ## Chain 1: Elapsed Time: 0.166227 seconds (Warm-up) ## Chain 1: 0.118388 seconds (Sampling) ## Chain 1: 0.284615 seconds (Total) ## Chain 1: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 2). ## Chain 2: ## Chain 2: Gradient evaluation took 1.4e-05 seconds ## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds. ## Chain 2: Adjust your expectations accordingly! ## Chain 2: ## Chain 2: ## Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 2: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 2: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 2: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 2: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 2: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 2: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 2: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 2: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 2: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 2: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 2: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 2: ## Chain 2: Elapsed Time: 0.151639 seconds (Warm-up) ## Chain 2: 0.120805 seconds (Sampling) ## Chain 2: 0.272444 seconds (Total) ## Chain 2: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 3). ## Chain 3: ## Chain 3: Gradient evaluation took 1.2e-05 seconds ## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds. ## Chain 3: Adjust your expectations accordingly! ## Chain 3: ## Chain 3: ## Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 3: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 3: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 3: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 3: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 3: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 3: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 3: ## Chain 3: Elapsed Time: 0.154105 seconds (Warm-up) ## Chain 3: 0.112509 seconds (Sampling) ## Chain 3: 0.266614 seconds (Total) ## Chain 3: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 4). ## Chain 4: ## Chain 4: Gradient evaluation took 1.2e-05 seconds ## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds. ## Chain 4: Adjust your expectations accordingly! ## Chain 4: ## Chain 4: ## Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 4: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 4: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 4: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 4: ## Chain 4: Elapsed Time: 0.166733 seconds (Warm-up) ## Chain 4: 0.1248 seconds (Sampling) ## Chain 4: 0.291533 seconds (Total) ## Chain 4:
## Inference for Stan model: poisson_regression. ## 4 chains, each with iter=2000; warmup=1000; thin=1; ## post-warmup draws per chain=1000, total post-warmup draws=4000. ## ## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat ## alpha 0.17 0 0.13 -0.09 0.09 0.18 0.27 0.43 785 1.01 ## beta 0.12 0 0.01 0.11 0.11 0.12 0.12 0.13 769 1.01 ## ## Samples were drawn using NUTS(diag_e) at Wed Jul 1 19:15:28 2020. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split chains (at ## convergence, Rhat=1).
Looking at Rhat we can see that we have reached the convergence.
theme_set(bayesplot::theme_default())
mcmc_scatter(as.matrix(fit.model.Poisson, pars=c("alpha", "beta") ), alpha=0.2)
Check the posterior:
y_rep <- as.matrix(fit.model.Poisson, pars="y_rep") ppc_dens_overlay(y = model.data$cases, y_rep[1:200,])
The model is not able to capture low and high numbers of new cases.
The fit is not satisfactory, it is probably due to overdispersion, we can check the residuals to confirm this hypothesis:
#in this way we check the standardized residuals mean_y_rep<-colMeans(y_rep) std_residual<-(model.data$cases - mean_y_rep) / sqrt(mean_y_rep) qplot(mean_y_rep, std_residual) + hline_at(2) + hline_at(-2)
The variance of the residuals increases as the predicted value increase. The standardized residuals should have mean 0 and standard deviation 1 (hence the lines at +2 and -2 indicates approximate 95% error bounds). The variance of the standardized residuals is much greater than 1, indicating a large amount of overdispersion.
Classically the problem of having overdispersed data is solved using the negative binomial model instead of the Poisson one.
In order to deal with the problem of overdispersed data, we can try to fit the following Negative Binomial model:
\[ ln\lambda_i = \alpha + \beta\cdot elapsed\_time_i \\ y_i \sim \mathcal{Negative Binomial}(\lambda_i, \phi) \]
Where the parameter \(\phi\) is called precision and it is such that:
\[ E[y_i] = \lambda \\ Var[y_i] = \lambda + \frac{\lambda^2}{\phi} \]